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\begin{document}

\bibliographystyle{plain}

\pagestyle{empty}
\begin{flushright}
{\sl AI-Modeling}
\end{flushright}
\vfill
\begin{center} 
\hrule
\hrule
\vspace{.05cm}
\hrule
\vspace{.25cm}
{\bf 
CIS 600 - Topics in AI: Modeling - Reality Rules (I). Problem 3, Page 41.
}
\vspace{.25cm}
\hrule
\vspace{.05cm}
\hrule
\hrule
\vspace{1cm}
Saleh Elmohamed\\
saleh@\{cis, npac\}.syr.edu\\   
\end{center}

\vspace{1.5cm}

\vfill
\centerline{\today}
\vfill

\newpage

\pagestyle{plain}

\section{Outline of the Problem}
\setcounter{page}{1}

Let:
\be{prob.3.1}
\Omega = \text{\{all cubic curves in one variable z\}}\;\;,
\ee
\be{prob.3.2}
\;\;\;\;\; = \{z^{3} + az + b: a, b real\}.
\ee

Consider two descriptions of $\Omega$ given by the observables.

{\bf\underline{A parameterizations description}}:
\be{obs.1.1}
f_1: \Omega \longrightarrow S  =  R^2\;\;,
\ee
\be{obs.1.2}
\omega \mapsto (a, b)
\ee

{\bf\underline{A coordinatizations description}}:
\be{obs.2.1}
f_2: \Omega \longrightarrow \{1,2,3\} = \hat{S}\;\;,
\ee
\be{obs.2.2}
\omega \mapsto \text{the number of real roots of} \omega\;.
\ee

{\bf\underline{Main Condition of the Problem}}:

A point $\alpha \in S$ (or $\hat{S}$) {\it generic} if there exists an
open neighborhood $U$ of $\alpha$ such that all $\hat{\alpha} \in U$
have the same root structure as $\alpha$; otherwise, $\alpha$ is a
{\it bifuraction point}.

{\bf\underline{Need to Show}}:

That the {\it generic points} under the description $f_1$ are
$\{(a,b): 4 a^3 + 27 b^2 \neq 0 \}$, while the {\it generic points}
under $f_2$ are $\{\O\}$, the empty set.

\section{Discussion, Definitions and Some Background}

From {\it example 2: The Cusp Catastrophe} we have the following
figure:

\vspace{7.0cm}

Showing the \underline{bifuraction set} of points on the curve
$4a^3 + 27b^2 = 0$. Also showing the cusp, a point at which two
branches of a curve meet, and at which the limits of the tangent on
each branch coincide. It is a {\bf cusp of the first kind} (or {\bf
simple cusp}) if the two branches are at opposite sides of the common
tangent.

Since $\Omega$ is the set of all cubic curves (represented by
third degree polynomials) --- algebraic curve of degree n --- the
$R^2$ in the above figure is partitioned into  3 disjoint classes.

\bd{neighborhood}
\underline{Neighborhood of a Point}: The $\epsilon$-neighborhood
of a point $\alpha$ is the set of all points whose distance from
$\alpha$ is less than $\epsilon$. A neighborhood of $\alpha$ is any
set that contains an $\epsilon$-neighborhood of $\alpha$, or, for
topological spaces, any set that contains an open set that contains 
$\alpha$. Sometimes neighborhood and open set are taken as
synonymous.
\ed

\bd{bifuraction-point}
A \underline{Bifuraction Point} is derived from
studying the behavior of a solution of a nonlinear problem in the {\bf
neighborhood} of a known solution, particularly as a parameter
varies. For more on these points, see topics of dynamic systems,
chaos, strange attractors, etc.

\ed
{\bf\underline{Example}}:
Given the following second order DE:
\be{DE.1}
y^{''} + (\lambda - {1 \over \pi} \int_0^{\pi} y^2\,dx) y = 0
\ee

has a trivial solution $y \equiv 0$. It has the solution $y = A
sin(nx)$ if $A^2 = 2(\lambda - n^2)$, where $A$ is a parameter. When
$\lambda > 1$ a new solution appears: $y = \sqrt{2(\lambda - 1)}
sin(x)$. When $\lambda \ge 4$, another solution $y = \sqrt{2(\lambda -
4)} sin(2x)$, etc. So the values 1, 4, 9, ... of $\lambda$ are {\bf
bifuraction points} or {\bf branching points} of the system.

\section*{A Rough Sketch of the Proof}

In the first description (eq. \ref{obs.1.1} and \ref{obs.1.2}), for
each point $\alpha \in S$, there is always another point $\hat{\alpha}$
that belongs to the same space $\hat{\alpha} \in S$ and also  happen to
fall in the $\epsilon-$neighborhood of $\alpha$. Furthermore, besides
having $\alpha$'s neighborhood to be open, it is also bounded by the
{\it cusp} curve $4 a^3 + 27 b^2 = 0$. Also, from the above Figure,
any $\epsilon-$neighborhood of $\alpha \in S$ will fall in one of the
regions: ``1'', ``2'', or ``3''. 

On the other hand, in the second description (eq. \ref{obs.2.1} and
\ref{obs.2.2}), we are dealing with a discrete space, so the only way
$\hat{\alpha} \in \hat{S}$ will fall in the same neighborhood as that of
$\alpha \in \hat{S}$ is if $\alpha = \hat{\alpha}$. That is, unless
$\alpha = \hat{\alpha}$, the two points won't have the same ``root
structure''. 

In summary, because of the discrete structure of $\hat{S}$, no {\it
generic} points set exists satisfying the main condition of the
problem. While for $S$, the set of {\it generic} points satisfying the
main condition consists of all non-bifuraction points or points that
don't fall on the {\it cusp} curve.


\end{document}








